This article will derive equations for converting a planet's ecliptic coordinates to equatorial coordinates. This conversion is necessary for calculating primary directions - the ancient technique of event prediction.
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Notations
First, let's define terms. We will denote
- Celestial longitude and latitude by $\lambda$ and $\delta$ letters
- Equatorial coordinates - right ascension and declination by $RA$ and $D$
- Ecliptic inclination by $\epsilon$
Coordinate rotation
Let's denote the vector (a pointer) to the observed planet with the letter $\vec{v}$. Then the spherical coordinates of this vector in the ecliptic coordinate system will be
$$ \vec{v} = (\lambda, \delta) $$
The Cartesian coordinates of the same vector, according to equation (2) will be equal to
$$ \begin{aligned} \vec{v} &= \cos\delta\cos\lambda~\vec{X_{ecl}} \\ & + \cos\delta\sin\lambda~\vec{Y_{ecl}} \\ & + \sin\delta~\vec{Z_{ecl}} \end{aligned} $$
Here we set $R$ (the radius of the celestial sphere) to be equal to 1 for simplicity. The $X_{ecl}$-axis directs to 0° Aries, the $Y_{elc}$-axis to 0° Cancer, and the $Z_{ecl}$-axis to the northern celestial hemisphere.
The equatorial plane is inclined at an angle $-\epsilon$ relative to the plane of the ecliptic in the $Y_{ecl}Z_{ecl}$ plane.
We can use the rotation matrix, which we introduced earlier, substituting ($-\sin\epsilon$) instead of $\sin(-\epsilon)$.
$$\mathbf{A}_{YZ} = \left[\begin{array} {rrr} 1 & 0 & 0 \\ 0 & \cos\epsilon & -\sin\epsilon \\ 0 & \sin\epsilon & \cos\epsilon \end{array}\right] $$
It gives us cartesian coordinates of the vector $\vec{v}$ in an equatorial coordinate system:
$$ \begin{cases} x_{eq} = \cos\delta\cos\lambda \\ y_{eq} = \cos\epsilon\cos\delta\sin\lambda - \sin\epsilon\sin\delta \\ z_{eq} = \sin\epsilon\cos\delta\sin\lambda + \cos\epsilon\sin\delta \end{cases} $$
From the conversion equation (1) it follows that
$$ \begin{cases} \tan(RA) = y_{eq} / x_{eq} \\ \sin(D) = z_{eq} \end{cases} $$
It gives us the final equations for converting $(\lambda, \delta)\rightarrow(RA, D)$:
Invert Conversion
For insvert conversion we change $\epsilon$ to $(-\epsilon)$. It gives us the following:
